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PI Passivity-Based Control : Application to PhysicalSystems

Rafael Cisneros Montoya

To cite this version:Rafael Cisneros Montoya. PI Passivity-Based Control : Application to Physical Systems. Automatic.Université Paris-Saclay, 2016. English. �NNT : 2016SACLS187�. �tel-01368308�

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Acknowledgments

I would like to express my gratitude to Prof. Ortega for his invaluable advices and

assistance during this period. He was an example of rigourous research and hard

work.

Thanks to Professors: R. Griñó, J. Scherpen, S. Aranovskiy and H. Mounier for

being part of the Jury.

I also want to thank to my parents and sister for beeing always there, supporting me.

To Sra. Amparo for her hospitality. To my laboratory collegues: Diego, Erik, Jonathan,

Lupe, Mattia, Missie, Mohamed, Pablo, Vı́ctor, for making this stay pleasant, creating

a friendly environment both inside and outside the laboratory.

Finally, thanks to the Consejo National de Ciencia y Tecnologı́a (CONACyT) for fund-

ing this thesis work.

3

Contents

Aperçu de la thèse 11

1 Introduction 15

1.1 Passivity : A control design tool . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Thesis Overview & Contributions . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 The Tracking PI–PBC : Application to Power Converters 23

2.1 The PI-PBC of Bilinear system for the regulation and tracking problem . 23

2.2 Passivity of the Bilinear Incremental Model . . . . . . . . . . . . . . . . . 25

2.3 A PI Global Regulating Controller . . . . . . . . . . . . . . . . . . . . . . 26

2.4 A PI Global Tracking Controller . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Application to Power Converters . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1 The Buck Converter . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.2 The Boost Converter . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Robust PI–PBC : An Application to Temperature Regulation 33

3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Robust PI control problem . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Assumptions on the open–loop plant . . . . . . . . . . . . . . . . 37

3.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Preliminary Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 The Robust PI–PBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Additional Remarks on the PI–PBC . . . . . . . . . . . . . . . . . . . . . . 43

5

6 CONTENTS

3.4.1 General G (with n−m zero rows) . . . . . . . . . . . . . . . . . . 43

3.4.2 Difficulties for adaptation . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.3 Comments on robustness based on continuity . . . . . . . . . . . 45

3.5 Application to Temperature Regulation . . . . . . . . . . . . . . . . . . . 45

3.5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5.2 Passivity of the thermal system. . . . . . . . . . . . . . . . . . . . 47

3.5.3 Robust PI–PBC of the thermal system . . . . . . . . . . . . . . . . 49

3.5.4 Numerical Simulation: . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Energy Shaping PI of PH Systems 53

4.1 Problem Formulation and Main Assumptions . . . . . . . . . . . . . . . . 53

4.2 Energy Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Relation with Classical PBCs . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.1 Energy–balancing PBC . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.2 Interconnection and damping assignment PBC . . . . . . . . . . . 58

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5.1 Micro electro–mechanical optical switch . . . . . . . . . . . . . . . 60

4.5.2 Two-Tanks Level Regulation Problem . . . . . . . . . . . . . . . . 62

4.5.3 LTI systems: Controllability is not enough . . . . . . . . . . . . . 63

5 Applications of the PI-PBC to Wind Energy Systems 67

5.1 PI–PBC of a Wind Energy System with Guaranteed Stability Properties . 68

5.1.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.2 Control Design : The PI-PBC Approach . . . . . . . . . . . . . . . 73

5.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 PBC of a Grid–Connected Small Scale Windmill . . . . . . . . . . . . . . 76

5.2.1 Mathematical Model of the System . . . . . . . . . . . . . . . . . . 78

5.2.2 Dynamics of the PMSG . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.3 Assignable Equilibria and Problem Formulation . . . . . . . . . . 81

5.2.4 A Cascade Decomposition of the System . . . . . . . . . . . . . . 83

CONTENTS 7

5.2.5 Standard PBC of Subsystem (5.45) . . . . . . . . . . . . . . . . . . 84

5.2.6 A Tracking PI PBC for Subsystem (5.46) . . . . . . . . . . . . . . . 91

5.2.7 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.8 Control Implementation . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.9 Benchmark system . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.10 Computer simulations . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Conclusions & Future Work 103

8 CONTENTS

Notations & Acronyms

Mathematical Notation

R Field of real numbers.

Rn Linear space of real vectors of dimension n.

Rn×m Ring of matrices of size n×m.

Rn≥0 For a vector x, xi ≥ 0, i = 1, . . . , n.

xi The i-th element of the vector x.

xij xij = col(xi, xi+1, . . . , xj) where i, j are integers such that 1 ≤ i < j ≤ n.

In The identity matrix of size n× n.

0n×s Matrix of zeros of n× s

0n column vector of zeros of dimension n.

diag(·) Diagonal matrix of the input arguments

col(·) Column vector of the input arguments

sym(·) Returns the symmetric part of a square matrix.

|x|2 Square of the Euclidean norm, i.e., |x|2 := x>x

‖x‖2S The weighted square Euclidean norm, i.e., ‖x‖2S := x>Sx.

g† Pseudo inverse of the full-rank matrix g, i.e., g† := (g>g)−1g>.

F?, F̃ (x) For the distinguished element x? ∈ Rn and any mapping F : Rn → Rs,

we denote F ∗ := F (x∗) and F̃ (x) := F (x)− F ∗.

g′, g′′ For mappings of scalar argument g : R→ Rs denote, respectively,

first and second order differentiation.

∇H(x) For H : Rn → R, it refers to the gradient operator of a function,

i.e.,∇H(x) :=(∂H(x)∂x

)>.

9

10

∇2H(x) For H : Rn → R, it refers to the Hessian operator of a function,

i.e.,∇2H(x) :=(∂2H(x)∂x2

)>.

∇C(x) For C : Rn → Rm, ∇C(x) = [∇C1(x), . . . ,∇Cm(x)].

arg max f(x) Returns the argument x of the maxima of a function f : Rn → R.

Acronyms

PBC Passivity-Based Control

SPBC Standard Passivity-Based Control

PMSG Permanent Magnet Synchronous Generator

AS Asymptotically Stable

UAS Uniformly Asymptotically Stable

GAS Global Asymptotically Stable

PI Proportional and Integral

PID Proportional-Integral-Derivative

PDE Partial Differential Equations

LMI Linear Matrix Inequalities

Aperçu de la thèse

Le régulateur PID (Proportionnel-Intégral-Dérivée) est la commande par retour d’état

la plus connue. Elle permet d’aborder un bon nombre de problèmes de commande,

particulièrement pour des systèmes faiblement non linéaires et dont la performance

requise est relativement modeste. En plus, en raison de sa simplicité, la commande

PID est largement utilisée dans l’industrie. Étant donné que les méthodes de réglage

de la commande PID sont basées sur la linéarisation, la synthèse d’une commande au-

tour d’un point d’équilibre est relativement simple, néanmoins, la performance sera

faible dans des modes de fonctionnement loin du point d’équilibre. Pour surmonter

ce désavantage, une pratique courante consiste à adapter les gains du PID, procédure

connue sous le nom de séquencement de gain (ou gain-scheduling en anglais). Il y a

plusieurs désavantages à cette procédure, comme la commutation des gains de la com-

mande et la définition –non triviale– des régions de l’espace d’état dans lesquelles cette

commutation aura lieu. Ces deux problèmes se compliquent quand la dynamique est

fortement non linéaire.

L’un des avantages à utiliser la passivité est son caractère intuitif, qui exploite les

propriétés physique des systèmes. Grosso modo, l’idée centrale d’un système pas-

sif est que le flux de puissance entrante au système n’est pas inférieur à l’incrément

de son énergie de stockage. Par conséquence, ces systèmes ne peuvent pas stocker

plus d’énergie que celle fournie, dont la différence correspond à l’énergie dissipée.

En introduisant le concept d’énergie, cette méthodologie nous permet de formuler le

problème de commande comme celui de trouver un système dynamique dont la fonc-

tion de stockage d’énergie prend la forme désirée. En incorporant le concept d’énergie,

cette méthode devient accessible à la communauté de praticiens et permet de fournir

des interprétations physiques de l’action de commande.

Dans ce contexte, ce travail de thèse a comme objectif de synthétiser des comman-

des PI, basées sur la passivité, de telle sorte que la stabilité globale du système en

boucle fermé soit garantie.

Ce travail de thèse est principalement la continuation de [16,23,32,35,65,69]. Parti-

culièrement dans [65], il est prouvé que si un système non-linéaire est rendu passif par

11

12

une loi de commande constante, il est stabilisable par une commande PI. En partant

de ce résultat, dans [32], une commande PI pour une classe de systèmes bilinéaires

est proposée pour traiter le problème de régulation et appliqué aux convertisseurs de

puissance. Dans cette thèse nous étendons ce dernier résultat au problème de suivi.

D’autre part, dans [35], des conditions suffisantes pour une classe de système dy-

namique sont énoncées telles que s’il est passif, sa représentation incrémentale l’est

aussi. Dans ce travail, en utilisant ce résultat, nous proposons une commande PI ro-

buste qui dépend seulement des paramètres de la matrice d’entrée.

Une contribution finale concerne le domaine de recherche développée par [23].

Nous proposons une méthodologie constructive pour la synthèse d’une commande

PI pour une classe de systèmes port-Hamiltoniens. Cette commande nous permet de

façonner l’énergie du système en boucle fermée.

Présentation de la thèse

La thèse est organisée de la manière suivante :

- Dans le Chapitre 2, nous présentons la commande PI-PBC qui adresse le problème

de suivi d’une classe de systèmes bilinéaires. Ce résultat est appliquée aux con-

vertisseurs de puissance qui sont décrits par les équations dynamiques de la

forme ẋ(t) = [A + ∑i ui(t)Bi]x(t) + B0u + d(t). L’approche est validée par dessimulations numériques.

- En Chapitre 3 nous étudions le problème de commande des systèmes non linaires

qui sont partiellement connus. Nous identifions une classe de systèmes dans

laquelle une commande PI basée sur la passivité peut stabiliser les système au-

tour d’un point d’équilibre désiré en connaissant seulement les paramètres de la

matrice d’entrée.

- Le Chapitre 4 propose une commande PI, complètement constructive, pour une

classe de systèmes porte-Hamiltoniens. En utilisant la sortie passive de façonnage

de puissance, cette commande nous permet de assigner, au système en boucle

fermée, le point minimum désiré de la fonction d’énergie du système en boucle

fermée.

- Le Chapitre 5 est consacré à des applications de la commande PI-PBC. En util-

isant la théorie des chapitres précédents, nous proposons deux systèmes d’énergie

éolienne et faisons la synthèse de sa commande. Cette commande a comme ob-

jectif de garantir la l’extraction de la puissance maximale provenant du vent.

Résumé 13

- Ce travail est terminé avec des conclusions et des travaux futurs, présentés dans

le Chapitre 6.

Publications

Ce travail de thèse a fait l’objet des articles de conférence et revue suivantes.

Articles de conférence

- R. Cisneros, M. Pirro, G. Bergna, R. Ortega, G. Ippoliti and M. Molinas, Global

Tracking Passivity-based PI Control of Bilinear Systems and its Application to

the Boost and Modular Multilevel Converters, Conf. on Modelling, Ident. and

Cont. of Non Syst. (MICNON), vol. 48, no. 11, pp. 420-425, 2015.

- S. Aranovskiy, R. Ortega and R. Cisneros, Robust PI passivity-based control of

nonlinear systems: Application to port-Hamiltonian systems and temperature

regulation, American Control Conference (ACC), pp. 434-439, 2015.

- P. Borja and R. Cisneros and R. Ortega, Shaping the energy of port-Hamiltonian

systems without solving PDE’s, 54th IEEE Conference on Decision and Control (CDC),

pp. 5713-5718, 2015.

- R. Cisneros, R. Gao, R. Ortega and I. Husain, PI Passivity–Based Control for

Maximum Power Extraction of a Wind Energy System with Guaranteed Stability

Properties, Int. Conf. on Renewable Energy : Generation and Appl. (ICREGA)(To be

printed), 2015.

Articles de revue

- R. Cisneros, F. Mancilla-David and R. Ortega,Passivity-Based Control of a Grid-

Connected Small-Scale Windmill With Limited Control Authority ,IEEE Journal

of Emerging and Selected Topics in Power Electronics, vol. 1, no. 4, pp. 247-259, 2013.


tracking passivity-based PI control of bilinear systems: Application to the inter-

leaved boost and modular multilevel converters, Control Engineering Practice, vol.

46, pp. 109-119, 2015.

- S. Aranovskiy, R. Ortega and R. Cisneros, A robust PI passivity-based control

of nonlinear systems and its application to temperature regulation, International

Journal of Robust and Nonlinear Control, vol. 26, no. 10, pp. 2216–2231, 2015.

14

- P. Borja, R. Cisneros and R. Ortega, A Constructive Procedure for Energy Shaping

of Port–Hamiltonian Systems, Automatica, vol. 72, pp. 230-234 , 2016.

.

Chapter 1

Introduction

Automatic feedback control systems have been known and used for more than 2000

years. There is evidence that the ancient romans and greeks developed devices to

regulate the water level [10]. In late medieval, attempts to provide speed regulation

by primitive feedback devices were made. Along the history, ingenious inventions

concerning feedback control systems have been reported: mechanisms to control tem-

perature, governors for steam engines, steering engines or servo-mechanisms, pneu-

matic feedback amplifiers, anti-aircraft control and many others. Actually, the word

feedback is a 20th century neologism introduced in the 1920s by radio engineers to

describe parasitic, positive feeding back of the signal from the output of an ampli-

fier to the input circuit. Interested reader is referred to [9, 10], where an interesting

monograph about the history of automatic control is presented.

The idea of feedback is at the same time, simple and powerful. It has had a pro-

found influence on technology. Application of the feedback principle has resulted in

major breakthroughs in control, communication, and instrumentation. The principle

of the (negative) feedback relies on increasing a manipulated variable (control input)

when the process variable is smaller than the setpoint (reference) and decrease the ma-

nipulated variable when the process variable is larger than the setpoint [6]. In general,

a process or system to be controlled is fed-back by a function of its measured signals.

One of the best known forms of feeding back a system is through a three-term con-

trol law known as PID (Proportional-Integral-Derivative) controller, which was firstly

presented with an analytical formalism by N. Minorsky. In a traditional scheme, an

error signal is derived from the difference between the measured signals and its de-

sired values. In the non-interacting PID, the control signal is based on a sum of the

weighted integral, proportional and derivative of the error. The transfer function of

15

16

this controller is of the form

HPID(s) = K(

1 + 1TIs

+ TDs).

The three term functionalities include:

1. The proportional term provides an overall control action proportional to the er-

ror signal through the allpass gain factor.

2. The integral term reduces steady-state errors through low-frequency compensa-

tion.

3. The derivative term improves transient response through high-frequency com-

pensation.

Other classical PID realization is the interacting PID, which is implemented as a cas-

cade of a PI and PD controller [6]. The transfer function of such PID is

HPID(s) = K(

1 + 1TIs

)(1 + sTd) .

Trying to improve transient performance has given rise to other control schemes such

as PI-D (type B) or I-PD (type C) controllers, see [3, 44] for a description.

PID controllers are sufficient for many control problems, particularly when process

dynamics are not highly nonlinear and the performance requirements are modest. Be-

sides, because of its simple structure, the PID controller is the most adopted control

scheme by industry and practitioners. However, many practitioners opt to switch off

the derivative term. Actually, many controllers applied in the industry are only PI

controllers [7].

An important issue when implementing a PID is to determine its parameters that

influence the performance of the system.In order to make this tuning of the PID gains

more constructive, some procedures have been appear in the literature. This methods

determine the gains values based on some parameters taken from the system response.

They are divided in frequency and step response methods. The first and more classical

method is the Ziegler-Nichols, which is frequently adopted because of its simplicity

to implement. In its step response version, two parameters are registered from the

straight line tangent at the inflection point of the step response of the system. Then,

the PI(D) controller parameters are obtained from a table. In its frequency response

version, the point at which the Nyquist curve intersects with the negative real axis is

determined. To do so, the process is fed back with a proportional controller, the pro-

portional gain is increased until the system and starts to oscillate. The proportional

Introduction 17

gain and the period of the oscillations are registered and using a table, the parameter

values of the PI(D) can be obtained. As discussed in [6], a fundamental drawback in

the Ziegler-Nichols Method is that the design criterion is focused in the decay ratio,

i.e., the ratio between two consecutive maxima of the error for a step change in set-

point or load. In this method, the closed–loop system has a quarter amplitude decay

ratio. This may cause good rejection of load disturbances but also poor damping and

stability margins. In order to improve the control performance, new tuning methods

have been developed [6]. In these methods the system response is characterized us-

ing three parameters instead of two, as in Ziegler-Nichols method. Even though these

methods improves substantially the performance, there is a trade-off between the sim-

plicity of the methods and its performance.

As stated in [6], many tuning strategies proposed can easily be eliminated if they

are compared with a well–tuned PID. Also, since these methods are based on the lin-

earization, commissioning a PI to operate around a single operating point is relatively

easy, however, the performance will be below par in wide operating regimes, which

is the scenario in modern high–performance applications. To overcome this draw-

back the current practice is to re–tune the gains of the PI controllers based on a linear

model of the plant evaluated at various operating points, a procedure known as gain–

scheduling. There are several disadvantages of gain–scheduling including the need

to switch (or interpolate) the controller gains and the non–trivial definition of the re-

gions in the plants state space where the switching takes place—both problems are

exacerbated if the dynamics of the plant is highly nonlinear. Another common com-

missioning procedure is to use auto–tuners, that heavily rely on the availability of a

“good” linear approximation of the plant dynamics. Besides, in other scenarios, a lit-

tle or no information about the dynamics of the process/system is known, thus no

stability of the system can be proved.

The current thesis work is aimed at the designing of PI controllers, based on the

passivity theory, such that the stability of the closed–loop system is guarantied. The

main objective is to develop constructive procedures that are applicable to physical

systems.

1.1 Passivity : A Control Design Tool [58, 63, 73].

Passivity concepts offer a physical and intuitive appeal. This is one of its main advan-

tage that explains the longevity of the concept from the time of its appearance —60

years ago. The primary idea in passive systems is that the power flowing into the sys-

tem is not less that the increase of storage. Thus, they cannot store more energy than

18

is supplied to it from the outside, with the difference being the dissipated energy.

It is clear from this energy interpretation that the concept of passivity is related

with the stability properties of the systems. For instance, rationalizing a feedback in-

terconnection as a process of energy exchange it is not surprising to learn that passivity

is invariant under negative feedback interconnection. In other words, the feedback in-

terconnection of two passive systems is still passive. If the overall energy balance is

positive, in the sense that the energy generated by one subsystem is dissipated by the

other one, the closed loop will be stable.

Viewing dynamics systems as energy-transformation devices is particularly useful

in studying complex nonlinear systems by decomposing them into simpler subsys-

tems that,upon interconnection, add up their energies to determine the full system’s

behavior. This allows to recast the control problem as finding a dynamical system

and an interconnection pattern such that the overall energy function takes the de-

sired form. This ”energy-shaping” approach is the essence of passivity-based control

(PBC). Moreover, because of the universality of the concepts of energy, this formula-

tion allows to facilitate the communication between practitioners and control theorists

incorporating prior knowledge and providing physical interpretations of the control

action.

The idea of energy shaping from a control point of view dates back to [77] where

a robot manipulator control methodology was proposed using this philosophy. Using

the fundamental notion of passivity, the principle was later formalized in [61], where

the term PBC was coined to define a controller design methodology whose aim is to

render the closed-loop system passive with a given storage function. Although there

are many variations of this basic idea, PBCs may be broadly classified into two large

groups, ”classical” PBC where we a priori select the storage function to be assigned

(typically quadratic in the increments) and then design the controller that renders the

storage function non-increasing. This approach, clearly reminiscent of standard Lya-

punov methods, has been very successful to control physical systems described by

Euler-Lagrange equations of motion, which as thoroughly detailed in, includes me-

chanical, electrical and electromechanical applications. Approaches within this cate-

gory are the energy balancing (EB), standard passivity based control (SPBC) [58] and

the PI-PBCs. In the second class of PBCs we do not fix the closed-loop storage function,

but instead select the desired structure of the closed- loop system, for example, La-

grangian or port-controlled Hamiltonian (PCH), and then characterize all assignable

energy functions compatible with this structure. This characterization is given in terms

of the solution of a partial differential equation (PDE). The most notable examples

of this approach are the controlled Lagrangian, the interconnection and damping as-

Introduction 19

signment (IDA) and power-shaping control methods which yields static controllers

(see [13, 29, 54] for further details). There has been reported a dynamic version of the

IDA-PBC, however as proved in [5], this extension is unnecessary since a system can

be stabilized by IDA-PBC if and only if it can be stabilized by the dynamic IDA-PBC.

In the same category is found the so-called control by interconnection (CBI). In this ap-

proach, a dynamic controller is obtained such that plant and the controller are coupled

via a power-preserving interconnection-generating an overall PCH system with stor-

age function the sum of the plant and controller storage functions. Roughly speaking,

in this methodology it is desired to achieve stabilization making the desired equilib-

rium a minimum of the new storage function [53].

The present thesis work is placed within this line of research. Particularly, we are

interested in deriving controllers, based on passivity theory, such that they admit a PI-

like structure. We called these controllers PI passivity-based controllers or simple PI-

PBC. The result presented is twofold. Firstly, we derived PI controllers that are widely

accepted by practitioners due to its simplicity and then we find application of these

controllers in physical systems. Secondly, the procedures here adopted encompass a

large class of nonlinear dynamic systems and do not need to solve PDEs, a situation

that commonly emerges when it is desired to shape the system energy.

1.2 Thesis Overview & Contributions

The work presented along this thesis follows mainly from [16, 23, 32, 35, 65, 69]. Thus,

the PI controllers here formulated extend the methodology presented therein. A brief

introduction of this line of research as well as the contributions of the thesis is de-

scribed in the following.

Borrowing the concepts of incremental passivity and energy in the increment of [85],

in [69] is shown that the energy in the increment of a broad class of switched converters

is a Lyapunov function for a given nominal trajectory, so the nominal trajectory is

stable. Furthermore, a control law of the incremental input is proposed to regulate

switched converters. This incremental input is a deviation of the control signal from

its value at the desired equilibrium (also known as nominal value). To represent the

original system control input from its increment, the knowledge of its nominal value

is needed.

In [65] is proved that if a nonlinear system is passifiable via a constant action, then

it is stabilizable with a PI controller depending on its passive output. Using this result,

in [32] is designed a completely constructive PI passivity-based controller for a class

of bilinear system and motivated by the application to power converters.

20

On the other hand, in [35] sufficient conditions are stated under which a class of

nonlinear systems defining a passive mapping u 7→ y, also defines a passive mappingin its incremental representation, i.e. the mapping ũ 7→ ỹ is passive —the symbol (̃)represents, respectively, the increments on the control input and output respect to their

value at the desired equilibrium point.

In the current thesis is presented an extension of the regulation case of power con-

verters system, addressed in [32], to the tracking problem. As a result, we prove that

the fulfillment of some given conditions makes possible to obtain —time-varying— PIs

controllers tracking admissible system trajectories. We apply this approach to some

benchmark examples and a wind energy system. Furthermore, experimental applica-

tions were reported in [21]. Another contribution of this thesis is intended to robustify

the PI-PBC controller introduced in [35]. To carry out this robustification we identify a

class of systems to which this technique is applicable when the system parameters are

unknown. This approach is motivated by its application to temperature regulation.

A final contribution of the thesis concerns the line of research developed in [23].

In that paper, it has been proposed for mechanical systems to abandon the objective

of structure preservation and attention has been concentrated on the energy shaping

objective only. That is, to look for a static state–feedback that stabilizes the desired

equilibrium assigning to the closed-loop a Lyapunov function of the same form as

the energy function of the open–loop system but with new, desired inertia matrix and

potential energy function. However, it was not required that the closed-loop system

is a mechanical system with this Lyapunov function qualifying as its energy function.

In this way, the need to solve the matching equations is avoided. Under the same

philosophy, we consider now the case of port–Hamiltonian (pH) systems. The starting

point of the design is the well–known power shaping output [55]. Then, we construct

the new storage function from the original energy function of the system and its power

shaping output. We find out that under some conditions, PI controllers that regulate

the system behavior and guarantee the stability of the system.

1.3 Outline of the Thesis

The current thesis is organized as follows.

- In Chapter 2, we present a PI-PBC for the tracking problem of a class of bilin-

ear systems. We extend the result reported in [32], where a PI depending on

a passive output is developed to solve the regulation problem in power con-

verters . The class of systems under consideration has the form ẋ(t) = [A +∑i ui(t)Bi]x(t) + B0u + d(t). A set of matrices {A,Bi} are identified, via a linear

Introduction 21

matrix inequalities, for which it is possible to ensure global tracking of (admis-

sible, differentiable) trajectories with a simple linear time–varying PI controller.

Instrumental to establish the result is the construction of an output signal with

respect to which the incremental model is passive. The result is illustrated by the

application to some conventional DC/DC converters.

- In Chapter 3, We deal with the problem of control of partially known nonlin-

ear systems, which have an open–loop stable equilibrium, but we would like to

add a PI controller to regulate its behavior around another operating point. We

identify a class of nonlinear systems for which a globally stable PI can be de-

signed knowing only the systems input matrix and measuring only the actuated

coordinates.

- In Chapter 4 a new, fully constructive, procedure to shape the energy for a class

of port–Hamiltonian systems that obviates the solution of partial differential

equations is proposed. Proceeding from the well–known passive, power shaping

output we propose a nonlinear static state–feedback that preserves passivity of

this output but with a new storage function. A suitable selection of a controller

gain makes this function positive definite, hence it is a suitable Lyapunov func-

tion for the closed–loop. The resulting controller may be interpreted as a classical

PI—connections with other standard passivity–based controllers are also identi-

fied.

- Chapter 5 is advocated to the application of the PI-PBC. This chapter is mainly

divided in two parts. First, we propose a maximum power extraction control

with a PI-PBC for a wind system consisting of a turbine, a permanent mag-

net synchronous generator (PMSG), a rectifier, a load and one constant volt-

age source, which is used to form the DC bus. We propose a linear PI con-

troller, based on passivity, whose stability is guaranteed under practically rea-

sonable assumptions. In the second part, we consider the problem of control-

ling small–scale wind turbines providing energy to the grid. In this section, the

overall system consists of a wind turbine plus a PMSG connected to a single–

phase ac grid through a passive rectifier, a boost converter and an inverter. We

present a high performance, nonlinear, passivity–based controller combining

two Passivity-Based Control techniques: the Standard PBC and the tracking PI-

PBC. Asymptotic convergence to the maximum power extraction point together

with regulation of the dc link voltage and grid power factor to their desired val-

ues is ensured. The performance of the proposed controllers is compared via

computer simulations.

22

- Finally, in Chapter 6, we wrap out the present thesis with some concluding re-

marks and possible future work in the same line of research.

Articles de conférence


Tracking Passivity-based PI Control of Bilinear Systems and its Application to

the Boost and Modular Multilevel Converters, Conf. on Modelling, Ident. and

Cont. of Non Syst. (MICNON), vol. 48, no. 11, pp. 420-425, 2015.

- S. Aranovskiy, R. Ortega and R. Cisneros, Robust PI passivity-based control of

nonlinear systems: Application to port-Hamiltonian systems and temperature

regulation, American Control Conference (ACC), pp. 434-439, 2015.

- P. Borja and R. Cisneros and R. Ortega, Shaping the energy of port-Hamiltonian

systems without solving PDE’s, 54th IEEE Conference on Decision and Control (CDC)

, pp. 5713-5718, 2015.

- R. Cisneros, R. Gao, R. Ortega and I. Husain, PI Passivity–Based Control for

Maximum Power Extraction of a Wind Energy System with Guaranteed Stability

Properties, Int. Conf. on Renewable Energy : Generation and Appl. (ICREGA)(To be

printed), 2015

Articles de revue

- R. Cisneros, F. Mancilla-David and R. Ortega,Passivity-Based Control of a Grid-

Connected Small-Scale Windmill With Limited Control Authority ,IEEE Journal

of Emerging and Selected Topics in Power Electronics, vol. 1, no. 4, pp. 247-259, 2013.


tracking passivity-based PI control of bilinear systems: Application to the inter-

leaved boost and modular multilevel converters, Control Engineering Practice, vol.

46, pp. 109-119, 2015.

- S. Aranovskiy, R. Ortega and R. Cisneros, A robust PI passivity-based control

of nonlinear systems and its application to temperature regulation, International

Journal of Robust and Nonlinear Control, vol. 26, no. 10, pp. 2216–2231, 2015.

- P. Borja, R. Cisneros and R. Ortega, A Constructive Procedure for Energy Shaping

of Port–Hamiltonian Systems, Automatica, vol. 72, pp. 230-234 , 2016.

Chapter 2

The Tracking PI–PBC of Bilinear

Systems : Application to Power

Converters

Bilinear systems are a class of nonlinear systems that describe a broad variety of phys-

ical and biological phenomena [51] serving, sometimes, as a natural simplification of

more complex nonlinear systems. In this chapter we propose a PI Passivity-Based

controller for the global tracking of admissible, differentiable trajectories of a class of

bilinear systems.

The objective of this chapter is to provide a theoretical framework—based on the prop-

erty of passivity [34,81] of the incremental model—to establish such a result. Our mo-

tivation to pursue a passivity framework is that it naturally leads to the design of PI

controllers, which are known to be simple, robust and widely accepted by practition-

ers. The result presented in this chapter is an extension to the problem of tracking

trajectories, of [32, 35] that treat the regulation case (see Section 2.3). The proposed

result is illustrated by an application to power converters.

2.1 The PI-PBC of Bilinear system for the regulation and

tracking problem

Consider the bilinear system1

ẋ = Ax+ d+m∑i=1

uiBix+B0u (2.1)

1For brevity, in the sequel the time argument is omitted from all signals.

23

24

where x ∈ Rn, d ∈ Rn are the state and the known time-varying signal vector, respec-tively, u ∈ Rm, m ≤ n, is the control vector, and A ∈ Rn×n, Bi ∈ Rn×n, B0 ∈ Rn×m arereal constant matrices.

We will say that a function x? : R+ → Rn is an admissible trajectory of (2.1), if it isdifferentiable, bounded and verifies

ẋ? =Ax? + d+m∑i=1

u?iBix? +B0u? (2.2)

for some bounded control signal u? : R+ → Rm.The global tracking problem is to find, if possible, a dynamic state–feedback con-

troller of the form

ż = F (x, x?, u?) (2.3)

u = H(x, x?, z, u?), (2.4)

where F : Rn × Rn × Rm → Rq, q ∈ Z+, and H : Rn × Rn × Rm → Rm, such that allsignals remain bounded and

limt→∞

[x(t)− x?(t)] = 0, (2.5)

for all initial conditions (x(0), z(0)) ∈ Rn × Rq and all admissible trajectories.We characterize a set of matrices {A,Bi} for which it is possible to solve the global

tracking problem with a simple linear time–varying PI controller. The class is identified

via the following LMI.

Assumption 2.1. ∃P ∈ Rn×n such that

P = P> > 0 (2.6)

sym(PA) ≤ 0 (2.7)

sym(PBi) = 0, (2.8)

where the operator sym : Rn×n → Rn×n computes the symmetric part of the matrix,that is

sym(PA) = 12(PA+ A>P ).

To simplify the notation in the sequel the positive semidefinite matrix has been de-

fined

Q := −sym(PA). (2.9)

The Tracking PI–PBC : Application to Power Converters 25

2.2 Passivity of the Bilinear Incremental Model

Instrumental to establish the main result of the paper is the following lemma.

Lemma 2.1. Consider the system (2.1) verifying the LMI of Assumption 2.1 and an

admissible trajectory x?. Define the incremental signals (̃·) := (·) − (·)?, and the m–dimensional output function

y := C(x?)x̃ (2.10)

where the map C : Rn → Rm×n is defined as

C(x?) :=

x>? B

>1

...

x>? B>m

+B>0P. (2.11)

The operator ũ 7→ y is passive with storage function

V (x̃) := 12 x̃>Px̃. (2.12)

Hence, it verifies the dissipation inequality

V̇ ≤ ũ>y.

Proof. Combining (2.1) and (3.30) yields

˙̃x =(A+m∑i=1

uiBi)x̃+m∑i=1

ũiBix? +B0ũ. (2.13)

Now, the time derivative of the storage function (2.12) along the trajectories of (2.13) is

V̇ (x̃) = x̃>P[(A+

m∑i=1

uiBi)x̃+m∑i=1

ũiBix? +B0ũ]

= −x̃>Qx̃+ x̃>P[m∑i=1

ũiBix? +B0ũ]

≤ x̃>P[m∑i=1

ũiBix? +B0ũ]

= x̃>P([B1x?| . . . |Bmx?

]+B0

)ũ

= y>ũ,

where (2.8) of Assumption 2.1 has been used to get the second identity, (2.7) for the

first inequality, (2.8) again for the third equation and (2.10) for the last identity. ��

26

Remark 2.1. A key step for the utilization of the previous result is the derivation of the

desired trajectories x? and their corresponding control signals u?, which satisfy (3.30).

As shown in the examples below this may prove to be a very complicated task and

some approximations may be needed to derive them. Indeed, it is shown in [52] that

even for the simple boost converter this task involves the search of a stable solution

of an Abel ordinary differential equation, whose only stable trajectory is known to be

highly sensitive to initial conditions.

2.3 A PI Global Regulating Controller [32]

In this section we recall the result reported in [32] about the regulation in systems of

the form (2.1). For this case, we consider d and x? constant vectors. Then, it can be

seen that (3.30) becomes an algebraic equation, i.e., x? is an admissible equilibrium

point satisfying

0 = Ax? + d+m∑i=1

u?iBix? +B0u? (2.14)

for some u? ∈ Rm.

Lemma 2.2. Consider the system (2.1) verifying Assumption 2.1 with d and x?, u? such

that (2.14). Then, the system (2.1) in closed–loop with the PI controller

ż =− y

u =−Kpy +Kiz(2.15)

with output y given in 2.10 andKp, Ki > 0. For all initial conditions (x(0), z(0)) ∈ Rn+m

the trajectories of the closed–loop system are bounded.

Proof: The reader is referred to [32].

2.4 A PI Global Tracking Controller

From Lemma 2.1 the next proposition follows immediately.

Proposition 2.1. Consider the system (2.1) verifying Assumption 2.1 and an admissi-

ble trajectory x? in closed loop with the PI controller

ż =− y

u =−Kpy +Kiz + u?(2.16)


with output (2.10), (4.12) and Kp > 0, Ki > 0. For all initial conditions (x(0), z(0)) ∈Rn × Rm the trajectories of the closed-loop system are bounded and

limt→∞

ya(t) = 0, (2.17)

where the augmented output ya : R+ → Rm+n is defined as

ya :=C(x?)Q

12

x̃,with Q

12 the square root of Q given in (2.9). Moreover, if

rank

C(x?)Q

12

= n, (2.18)then state global tracking is achieved, i.e., (2.5) holds.

Proof. The PI controller (2.16) is equivalent to

ũ =−Kpy +Kiz

ż =− y.(2.19)

Consider the following radially unbounded Lyapunov function candidate

W (x̃, z) := V (x̃) + 12z>Kiz, (2.20)

whose time derivative is

Ẇ = −x̃>Qx̃+ y>ũ− z>Kiy

= −x̃>Qx̃− y>Kpy

≤ −λmin{Kp}|y|2 − |Q12 x̃|2 ≤ 0.

Notice that the closed-loop system (2.13) and (2.19) is non-autonomous because of its

dependence on u? and x? which are time-varying signals. Consequently, we cannot

invoke LaSalle’s Invariance Principle and proceed, instead, applying the generaliza-

tion of Barbalat’s Lemma reported in [79] and some standard signal chasing. Invoking

the aforementioned result, we must prove that ya ∈ L2 and ẏa ∈ L∞ to conclude thatlimt→∞ ya(t) = 0. Since the derivative of the Lyapunov function is negative, the trajec-tories are bounded, namely, z, x̃ ∈ L∞. In the same way, from the last inequality weconclude that y,Q

12 x̃ ∈ L2, consequently ya ∈ L2. To conclude that ẏa ∈ L∞ first notice

28

that x̃, x? ∈ L∞ implies x ∈ L∞ and, this in its turn, implies from (2.10) that y ∈ L∞.Now, y, z, u? ∈ L∞ implies, from (2.16), u ∈ L∞. That implies, from (2.13), ˙̃x ∈ L∞.Now, compute

ẏ =

ẋ>? B

>1

...

ẋ>? B>m

Px̃+x>? B

>1

...

x>? B>m

+B>0P ˙̃x, (2.21)

which is bounded because ẋ? ∈ L∞. Then, ẏa is bounded and it follows that ya(t)→ 0.

The proof of global state tracking follows noting that ya(t) → 0 ensures (2.5) if therank condition (2.18) holds. ��

Remark 2.2. Notice that the matrix C depends on the reference trajectory. Therefore,the rank condition (2.18) identifies a class of trajectories for which global tracking is

ensured.

Remark 2.3. To assess the effect of the approximations mentioned in Remark 2.1, con-

sider the following scenario where m = 1 and B0 = 0. Defining x̄ = x? + ξ andū = u? + ς as, respectively, the approximation of x? and u? with ξ and ς two boundedsignals, the measurable output is

ȳ = (x? + ξ)>B>Px

= y + ξ>B>Px.

Then, in this setting, the controller (2.19) becomes

ż = −y + ξ>B>P (x̃+ x?)

ũ = −Kpy +Kiz −Kpξ>B>Px+ ς.

Furthermore, the derivative of (2.20) yields

Ẇ = −x̃>Qx̃− y>Kpy + y>(ς −Kpξ>B>Px) + z>Kiξ>B>P (x̃+ x?).

Hence, from the latter, we cannot conclude stability since (2.20) is not a strict Lyapunov

function and there is no way to dominate the new terms appearing in its derivative.

Notice that for simplicity we adopt the case m = 1 however, it can be readily ex-tended for m ≥ 2.


C

+vE

+ i

L

R−

u

Figure 2.1: Representation of the ideal Buck Converter.x1,x1⋆

0

0.5

1

x2,x2⋆

0

0.5

1

x1⋆, x2⋆ x1, x2, u

5sec

div

u

0

0.5

1

Figure 2.2: Simulation result of the tracking PI-PBC for the Buck Converter

2.5 Application to Power Converters

The present section is intended to exemplify the use approach proposed within this

chapter. Experimental results have been reported in [21].

2.5.1 The Buck Converter

Consider the well-known normalized average model of the Buck Converter depicted

in Fig. 2.1:ẋ1 =− x2 + u

ẋ2 =x1 −x2D,

(2.22)

where D := R√

CL

. Also, E,C, L and R are the system parameters and u the control

input. The relation between the physical variables i, v and x is given by the following

transformation x1x2

= 1E√LC 0

0 1E

iv

. (2.23)

30

Clearly, defining

A =0 −1

1 − 1D

, B0 =1

0

, Q = P = I2, (2.24)the system satisfies Assumption 2.1. Furthermore,

y = B>0 Px̃ = x̃1. (2.25)

The control objective is to drive x2 to a desired time-varying reference x2?. Thus, from

second equation of (2.22),

x1? =ẋ2? +x2?D, (2.26)

which substituted in the first equation of (2.22) yields

u? = ẍ2? +ẋ2?D

+ x2?. (2.27)

Considering x2? = V0 + a sin(ωt), (2.26) and (2.27) becomes

u? =− aω2 sin(ωt) +a

Dω cos(ωt) + a sin(ωt) + V0

x1? =aω cos(ωt) +a

Dsin(ωt) + V0

D.

Fig. 2.2 shows the simulation results of the system (2.22) in closed–loop with the

controller (2.16). The system parameters are [75]: R = 25 Ω, C = 50 µC, L =19.91 mH, E = 24 V. Also, we select a = 0.3, ω = 0.8, V0 = 0.6, and gains Kp =0.3, Ki = 0.1.

2.5.2 The Boost Converter

The well-known normalized average model of the Boost shown in Fig. 2.3 is

ẋ1 =− x2u+ 1

ẋ2 =x1u−x2D

(2.28)

where D := R√

CL

. Also, E,C, L and R are the system parameters and u the control

input. The relation between the physical variables i, v and x is given by (2.23). Clearly,


C+v

E

+i

R−

uL

Figure 2.3: Representation of the ideal Boost Converter.

x1,x1⋆

0

4

8

x2,x2⋆

0

2

4

x1, x2, u x1⋆, x2⋆

17.5sec

div

u

0

0.5

1

Figure 2.4: Simulation result of the tracking PI-PBC for the Boost Converter.

defining

A =0 0

0 − 1D

, B1 =0 −1

1 0

, Q = P = I2, (2.29)the system satisfies Assumption 2.1. Furthermore,

y = x>? B>1 Px̃ = x̃2x1? − x̃1x2?. (2.30)

The control objective is x2, which is selected as x2? = V0 +a sin(ωt). On the other hand,from the second equation of (2.28) we have

u? =1x1?

(ẋ2? +

x2?D

). (2.31)

Substituting the latter equation in the first equation of (2.28) yields

ẋ1?x1? = x1? − x2?(ẋ2? +

x2?D

). (2.32)

As claimed in Remark 2.1, since the system contains only one stable solution, finding

x1? from (2.32) is a difficult task. Instead, we take the approximation of the solution of

32

such system proposed in [27]. Here below we write the expression of x̂1?, the approxi-

mation of x1? —refer [27] for further details:

x̂1? = c0 +1

2ωc0

[8V0aD

cos(ωt)− 4V0aω sin(ωt) + a2ω cos(2ωt) +a2

Dsin(2ωt)

](2.33)

where c0 := 1D(V 20 + a

2

2

).

Under this approximation, signal û?, the approximation of (2.31), becomes

û? =1x̂1?

(cos(ωt) + 1Da sin(ωt) + V0) (2.34)

In Fig. 2.4 the simulation plots are shown for the system (2.28) in closed–loop with

(2.16), when L = 18mH, C = 220µC, E = 50 V0 = 135 V, a = 15, ω = 0.6252 Kp =Ki = 0.5. A close-up view of the plot must reveal a steady-state error due to theapproximation of x1?.

Chapter 3

A Robust PI Passivity–Based Control of

a class of Nonlinear Systems :

Application to Temperature Regulation

In many practical applications the plant to be controlled has an open–loop stable equi-

librium, e.g., at the origin, and we would like to add a controller to regulate its behavior

around another operating point. In the case of linear systems the dynamics remains

invariant under coordinate shifts, therefore this task can be easily accomplished using

the incremental model of the plant. Unfortunately, this is not the case for nonlinear

systems, for which there is no obvious advantage of working with the incremental

model. To carry out this regulation task, in this chapter we identify a class of (input

affine) nonlinear systems for which it is possible to design a PI controller with the

following features.

F1 Regulation of the closed–loop system around the desired (non–zero) operating

point should be guaranteed.

F2 The PI controller should be robust, in the sense that reduced knowledge of the

system parameters is required.

F3 To simplify the controllers commissioning, a well defined admissible range of

variation for the PI proportional and integral gains, preserving closed–loop sta-

bility, should be provided.

We propose the construction of a PI controller with the features F1–F3 for plants with

unknown dynamics verifying the following assumptions.

A1 The open–loop system is unknown but has a stable equilibrium at the origin.

33

34

A2 The desired equilibrium belongs to the assignable set and admits a convex Lya-

punov function.

A3 The Lyapunov function is the sum of two functions, depending on the un–actuated

and actuated coordinates, respectively. The first function is unknown while the

second one is separable and linearly parameterized in terms of some unknown

parameters.

A4 The input matrix is constant, known and has n−m zero rows, where n and m arethe dimensions of the state and input vectors, respectively.

As indicated in the article’s title we exploit the fundamental property of passivity to

design the PI, which will be referred in the sequel as PI Passivity–based Control (PI–

PBC). The first step in the design is to, relying on A1 above, invoke the celebrated

theorem of Hill and Moylan [81] to identify a suitable passive output for the system,

with storage function the Lyapunov function of the open–loop system. Since our in-

terest is the regulation of non–zero equilibria, we then use the results of [35] to create

a new passive output for the incremental model with a storage function that has a

minimum at the desired equilibrium. As shown in [35], feeding back the passive out-

put through a PI controller ensures stability of the desired equilibrium for all positive

definite PI gains. It is important to underscore that, since the passivity property has

been established for the incremental model, the equilibrium can also be stabilized set-

ting the control input equal to the (constant) value that assigns the equilibrium, say u?,

which is univocally defined. However, this open–loop control will, obviously, be non–

robust. In the robustness context of the present chapter neither the plant dynamics nor

the Lyapunov function are known and, consequently, we cannot compute neither the

passive output nor u?. It is at this point that we invoke A3 and A4 above to prove that,

under these assumptions, it is possible to define suitable proportional and integral

gains that make the PI–PBC implementable and, consequently, guarantee stability of

the equilibrium. Another important feature of the proposed PI–PBC is that it requires

only partial measurement of the state, namely, only the m state variables associated to

the non–zero rows of the input matrix, referred in the sequel as actuated coordinates. In

this way, our approach is oriented towards a characterization of a class of systems that

can be regulated by means of the PI–PBC with a minimum knowledge of the system

parameters.

Several practical applications of PI–PBC have been reported in the literature. This

include, RLC circuits [16], power converters [32], fuel cells [78], electric drives [47] and

mechanical systems [49]. In [22] a procedure to add an integral action to a non–passive

output for a class of port–Hamiltonian systems was first proposed, and later extended

Robust PI–PBC : An Application to Temperature Regulation 35

in [60], [68]. To the best of our knowledge, the present result is the first attempt to

design PI–PBCs with guaranteed stability properties for systems with partially known

dynamics.

A natural question that arises at this point is the incorporation of adaptation in

the design of the PI (or PID). In the power converter application of [32] a parameter

that enters in the definition of the passive output, i.e., the load resistance, is adaptively

identified—however, all other parameters are assumed to be known. In the interesting

paper [4] it is shown that it is possible to adaptively estimate u? for a general nonlin-

ear system with scalar input, keeping the estimate in a known interval, provided the

passive output is known. In spite of a large number of publications the problem of

designing a provably stable adaptive PID for systems with unknown parameters re-

mains, as far as we know, open. The difficulty of this task was identified already in

1984 in [56]. As is well–known [70], the stability of indirect adaptive methods relies

on parameter convergence that, in its turn, requires persistency of excitation—a prop-

erty that is not satisfied in the regulation tasks where PI control is effective. On the

other hand, the application of direct methods is stymied by the absence of a suitable

parameterization of this structure–constrained controller. For the PI–PBC studied in

this chapter the main difficulty is the need to estimate two objects, that appear multi-

plicatively in the Lyapunov analysis: the passive output and the ideal control signal

u?. This point is further elaborated in Subsection 5.2.5.

3.1 Problem Formulation

In this section we formulate the control problem addressed along the chapter, enun-

ciate the assumptions made on the plant to solve it and make some remarks on these

assumptions.

3.1.1 Robust PI control problem

Consider the nonlinear, input affine, system

ẋ = f(x) +Gu, (3.1)

where x ∈ Rn, u ∈ Rm, n > m, f : Rn → Rn is an unknown smooth mapping, G ∈ Rn×m

is constant verifying rank(G) = m.The following is a key assumption made throughout the chapter.

36

Assumption 3.1. The matrix G has n − m zero rows. Without lost of generality1 it isassumed of the form

G =0(n−m)×m

G2

, (3.2)where G2 ∈ Rm×m is known.

This assumption can be easily obviated introducing state and input changes of coor-

dinates. Indeed, it is well–known—see, e.g., Theorem 2 of Section 2.7 of [43]—that for

any full rank, matrix G ∈ Rn×m there exists (elementary) full rank matrices T ∈ Rn×n

and S ∈ Rm×m such that

TGS =0(n−m)×m

Im

.Consequently, introducing z = Tx and v = S−1u the system (3.1) takes the desiredform

ż = w(z) +0(n−m)×m

Im

v,where w(z) = Tf(T−1z). We should note, however, that a change of state representa-tion destroys—in general—the original structure of the system, whose knowledge may

be critical for the verification of the second assumption below. This fact is clearly illus-

trated in the physical examples considered in Section 5.2.6. For this reason, we prefer

to leave it as an standing assumption.

Motivated by Assumption 3.1 we find convenient to define a partition of the state

vector into its un–actuated and actuated components as

x = xuxa

, xu :=

x1

x2...

xn−m

, xa :=

xn−m+1

xn−m+2...

xn

.

It is assumed that only xa is available for measurement.

The unforced system, that is, ẋ = f(x), has a stable equilibrium at the origin witha partially known Lyapunov function. We are interested in controlling the system with

a PI at a non–zero equilibrium—a situation that arises in most practical applications.

Thus, we are given a desired equilibrium point, x? ∈ Rn, and the control goal is to

1See R6 in the next subsection and Subsection 5.2.5 for more general forms of G.


ensure stability of this equilibrium using a PI control law of the form

ż = −KIψ(xa, x?)

u = −KPψ(xa, x?) + z

where z ∈ Rm is the controller state, KP ∈ Rm×m and KI ∈ Rm×m are tuning gainsand ψ : Rm × Rn → Rm is a mapping designed with the partial knowledge of theaforementioned Lyapunov function.

The following, practically reasonable, assumption is made throughout the chapter.

Assumption 3.2. The desired equilibrium point x? belongs to the assignable equilib-

rium set, that is,

x? ∈ E :={x ∈ Rn |

[In−m | 0(n−m)×n

]f(x) = 0

}. (3.3)

3.1.2 Assumptions on the open–loop plant

The following assumption identifies the class of vector fields f(x) for which we pro-vide an answer to the problem.

Assumption 3.3. For the system (3.1) there exists a twice–differentiable, positive defi-

nite function H : Rn → R≥0, verifying the following.

(i) [∇H(x)]>f(x) ≤ 0.

(ii) [∇H(x)−∇H(x?)]>f̃(x) =: −Q(x) ≤ 0.

(iii) The function H(x) is of the form

H(x) = Hu(xu) +Ha(xa) (3.4)

with

Ha(xa) =n∑

i=n−m+1diφi(xi), (3.5)

where the function Hu : Rn−m → R and the constants di > 0 are unknown but thefunctions φi : R→ R are known.

(iv) The functions Hu(xu) and φi(xi) are convex.

38

3.1.3 Discussion

The following remarks regarding the assumptions are in order.

R1 Although the set of assignable equilibria E is not known, it is reasonable to as-sume that we have enough prior knowledge about the plant to select the desired

operating point as a feasible equilibrium. Hence, Assumption 3.2 is practically

reasonable.

R2 A corollary of Assumption 3.2 is that the constant input u?, that assigns the equi-

librium, is uniquely defined as

u? :=(G>2 G2

)−1 [0m×(n−m) G>2

]f ?. (3.6)

Notice that, without knowledge of f(x), this constant cannot be computed.

R3 Since the open–loop system (3.1) has a stable equilibrium at the origin Assump-

tion 3.3 (i) follows as a corollary of Lyapunov’s converse theorems [39]. As will

become clear below Assumption 3.3 (ii) and (iv) are required to prove passivity

of the incremental model as done in [35].

R4 We underscore that no assumption, beyond twice differentiability and convex-

ity, is imposed on the unknown component Hu(xu) of the Lyapunov function ofthe open–loop system H(x). On the other hand, stricter conditions are imposedon the second component Ha(xa), with uncertainty captured by the unknownconstants di.

R5 Assumptions 3.3 (iii) and Assumption 3.1 are the key requirements imposed on

the plant to design the robust PI–PBC. This assumption is satisfied by a large

class of physical systems, including the thermal systems studied in Section 5.2.6

and a class of port–Hamiltonian systems [81].

R6 It can be noticed that the class of port-Hamiltonian systems of the form:

ẋ = (J −R)∇H(x) +Gu (3.7)

with constant interconnection J = −J > and damping R = R> ≥ 0 matricessatisfies Assumption 3.3 (i) and (ii). Indeed, Assumption 3.3 (i) is satisfied since

[∇H(x)]>(J −R)∇H(x) = −[∇H(x)]>R∇H(x) ≤ 0.


Similarly, Assumption 3.3 (ii) e

[∇H(x)−∇H(x?)]>(J −R)[∇H(x)−∇H(x?)] =

−[∇H(x)−∇H(x?)]>R[∇H(x)−∇H(x?)] ≤ 0.

R7 Regarding Assumptions 3.1, in the more general case when G is not of the form

(3.2) an additional shuffling of the rows of G is needed in the design. This proce-

dure is explained in Subsection 5.2.5.

R8 For quadratic Lyapunov functions of the form H(x) = x>Px, with P > 0, As-sumption 3.3 (ii) is satisfied if the open–loop system is convergent in the sense of

Demidovich [64]. That is, if it satisfies

P∇f(x) + [∇f(x)]>P ≤ 0.

3.2 Preliminary Lemmata

Unless otherwise indicated, throughout the rest of the chapter Assumption 3.1 holds.

Define for the system (3.1) the output

y = G>∇H(x) = G>2 DΦ(xa), (3.8)

where

D :=

dn−m+1 0 . . . 00 dn−m+2 . . . 0...

......

...

0 0 . . . dn

> 0

Φ(xa) :=

φ′n−m+1(xn−m+1)

...

φ′(xn)

.

A corollary of the theorem of Hill and Moylan [81] is that, if Assumption 3.3 (i) holds,

the system (3.1), (3.8) defines a passive mapping u 7→ y with storage function H(x).To operate the system at a non–zero equilibrium it is necessary to shift the mini-

mum of the storage function and define the passivity property between the incremen-

tal input and the output error. Towards this end, we recall Proposition 1 of [35] and

state it as a lemma below.

40

Lemma 3.1. Consider the incremental model of the system (3.1), (3.8)

ẋ = f(x) +Gu? +Gũ,

e = G>2 DΦ̃(xa),(3.9)

where ũ = u − u? is the incremental input. Under Assumptions 3.1–3.3 the mappingũ 7→ e is passive with storage function U : Rn → R≥0 given by

U(x) = H(x)− x>u∇Hu? − x>aDΦ? + k, (3.10)

where k is a constant that ensures U(x?) = 0. More precisely,

U̇ = −Q(x) + e>ũ, (3.11)

where Q(x) is defined in Assumption 3.3 (ii).

One of the main interests of passive systems is that they can be globally stabilized

with PI controls (with arbitrary positive definite gains). This well–known fact is stated

in the lemma below, whose proof is given to streamline the presentation of our main

result.

Lemma 3.2. Consider the system (3.1) verifying Assumptions 3.1–3.3 in closed–loop

with the PI–PBCe = G>2 DΦ̃(xa)

ż = −KIe

u = −KP e+ z.

(3.12)

For all positive definite gain matrices KP ∈ Rm×m and KI ∈ Rm×m all trajectoriesare bounded, the equilibrium point (x, z) = (x?, u?) is globally stable (in the sense ofLyapunov) and the augmented error signal

ea := Q(x)

e

(3.13)where Q(x) is defined in Assumption 3.3 (ii), verifies

limt→∞

ea(t) = 0. (3.14)

Moreover, if ea is a detectable output for the closed–loop system then the equilibrium

point is asymptotically stable.

Proof. Defining z̃ := z−u? the last two equations of the controller (3.12) may be written


in the form˙̃z = −KIe

ũ = −KP e+ z̃.(3.15)

Consider the Lyapunov function candidate

W (z̃, x) = U(x) + 12 z̃>ΛI z̃, (3.16)

where ΛI > 0. The time derivative of the Lyapunov function along the trajectories ofthe closed–loop system is

Ẇ = −Q(x) + e>ũ+ z̃>ΛI ˙̃z

= −Q(x)− e>KP e+ z̃>e− z̃>ΛIKIe.(3.17)

Setting ΛI = K−1I yieldsẆ = −Q(x)− e>KP e.

The proof is complete invoking standard Lyapunov arguments [39]. ��

3.3 The Robust PI–PBC

As indicated in R4 of Subsection 5.2.2 the matrixD is unknown. Hence, the error signal

e cannot be constructed and the PI–PBC (3.12) is not implementable. This motivates

our main result given below.

Proposition 3.1. Consider system (3.1) verifying Assumptions 3.1–3.3 in closed–loop

with the robust PI–PBCu = −KP Φ̃(xa) + z

ż = −KIΦ̃(xa),(3.18)

with the controller gains

KP = G−12 ΓPKI = G−12 ΓI . (3.19)

For all diagonal, positive definite matrices ΓP ∈ Rm×m and ΓI ∈ Rm×m we have thefollowing.

(i) All trajectories are bounded and the equilibrium point (x, z) = (x?, u?) is globallystable (in the sense of Lyapunov).

(ii) The augmented error signal ea defined in (3.13) verifies (3.14).

42

(iii) If ea is a detectable output for the closed–loop system then the equilibrium point

is globally asymptotically stable.

Proof. Some simple manipulations prove that

KP Φ̃(xa) = G−12 ΓPD−1G−>2 G>2 DΦ̃(xa) = ΛP e, (3.20)

where we defined the matrix

ΛP := G−12 ΓPD−1G−>2 , (3.21)

and used the definition of e in (3.12). Invoking Sylvester’s Law of Inertia [43], and the

fact that ΓP and D are diagonal and positive definite, we have that ΛP > 0.Next choose

ΛI := G>2 DΓ−1I G2, (3.22)

that is, also, positive definite for all diagonal, positive definite ΓI . Then

ΛIKIΦ̃(x) = G>2 DΦ̃(xa) = e. (3.23)

Replacing (3.20) and (3.23) in the controller equations yields

ũ = −ΛP e+ z̃˙̃z = −Λ−1I e.

Consequently, the time derivative of the Lyapunov function (3.17) becomes now

Ẇ = −Q(x)− e>ΛP e, (3.24)

completing the proof. ��

To obtain an implementable version of the robust PI–PBC it was necessary to carry–

out two tasks. First, to make the damping injection, introduced by the proportional

term, function of the unknown matrix D. Indeed, replacing (3.21) in (3.20) we get

KP Φ̃(x) = G−12 ΓPD−1G−>2 e.

Second, make the gain ΛI of the Lyapunov function (5.54) also a function of D—see(3.22).

An important observation is that, even though the controller only requires mea-

surement of the actuated terms of the state xa, it achieves regulation of the full state


vector.

3.4 Additional Remarks on the PI–PBC

In this section we explain how to proceed when G is not of the form (3.2), discuss the

reasons that stymie the use of adaptation and the inability to state a robustness result

based on continuity and approximate prior knowledge of the matrix D.

3.4.1 General G (with n−m zero rows)

Instrumental to design the robust PI–PBC was the particular form of H(x) defined inAssumption 3.3 (iii). In view of the construction of the robust PI–PBC, it is clear that

if G is not of the form (3.2) the assumption must be modified redefining the actuated

and un–actuated coordinates.

To avoid cluttering the notation we will explain the procedure only for the case

when n = 3 and m = 2—the general case follows verbatim. Assume, furthermore, thatG is of the form

G =

g>1

01×2g>3

.The form of H(x) given in Assumption 3.3 (iii) must be, accordingly, modified to

H(x) = Hu(x2) + d1φ(x1) + d3φ(x3).

In this case the passive output e for the incremental model becomes

G>[∇H(x)−∇H(x?)] = Gs

d1 00 d3

Φ̃1(x1)Φ̃3(x3)

.where

Gs :=[g1 | g3

].

The robust PI–PBC is given by

u = −G−1s ΓP

Φ̃1(x1)Φ̃3(x3)

+ zż = −G−1s ΓI

Φ̃1(x1)Φ̃3(x3)

,

44

where ΓP and ΓI are arbitrary, diagonal, positive definite matrices.Before closing this subsection we remark that our construction critically relies on

the assumption of existence of n−m zero rows in G. Indeed, it is possible to show thatif this is not the case, even assuming H(x) of the form

H(x) =n∑i=1

diφi(xi)

it is not possible to find an m × m positive definite matrix Λ, which will depend onD, such that the matrix ΛG>D is independent of D. The fact that this is not possible forall matrices G is obvious considering the counterexample G = col(1, 1). Hence, theassumption of existence of n−m zero rows in G is necessary to solve the problem.

3.4.2 Difficulties for adaptation

A natural alternative to the robust PI–PBC presented above is to assume a parametri-

sation of f(x) and try to estimate this parameters or, in a direct approach, estimate thematrix D that defines the passive output. The indirect approach, as is well–known, re-

lies on parameter convergence that requires persistency of excitation—a property that

is not satisfied in the regulation tasks where PI control is effective.

Let us see what are the difficulties for the application of a direct adaptation ap-

proach. Towards this end, we propose the adaptive PI–PBC

˙̂D = F (x, z)

ê = G>2 D̂ Φ̃(xa)

ż = −KI ê

u = −KP ê+ z,

where the parameter adaptation law F : Rn × Rm → Rm×m is to be defined.2 Definingẽ := ê− e the last two equations of the controller may be written in the form

˙̃z = −KI(e+ ẽ)

ũ = −KP (e+ ẽ) + z̃.

The time derivative of the Lyapunov function (5.54) with ΛI = K−1I is now

Ẇ = −Q(x)− ê>KP ê− ũ>ẽ

= −Q(x)− ê>KP ê− ũ>G>2 D̃Φ̃(xa)

2Notice that, in contrast to the robust PI–PBC, we have assumed that the full state is measurable.


where we underscore the presence of the last right hand term. If ũ were known the

standard procedure of augmenting the Lyapunov function with a term trace(D̃>D̃)and cancelling the sign–indefinite term with a suitable choice of F (x, z) would do thejob. Alas, u? is not known, hence this approach is not feasible.

Adding an adaptation for the constant u? is also not a trivial task, because of the

bilinear nature of the joint estimation problem.

3.4.3 Comments on robustness based on continuity

The availability of a bona fide Lyapunov function for the known parameters PI–PBC,

i.e., W (x, z̃), suggests that stability will be preserved if the matrix D is replaced by a“good”, constant estimate of it, say D0. More precisely, it is expected that replacing the

controller (3.12) bye0 = G>2 D0Φ̃(xa)

ż = −KIe0u = −KP e0 + z,

where

D = D0 + ∆, ∆ := diag{δi}

would ensure stability if |col(δi)| is sufficiently small. Unfortunately, since the Lya-punov function is not strict, this conjecture cannot be analytically validated. Indeed, in

this case the time derivative of the Lyapunov function (5.54) with ΛI = K−1I is now

Ẇ = −Q(x) + e>ũ− z̃>(e−G>2 ∆Φ̃(xa))

= −Q(x)− e>0 KP e0 − (KP e0 − z̃)>G>2 ∆Φ̃(xa).

While the term e>0 KPG>2 ∆Φ̃(xa) can be dominated for “small” ∆, there is no way wecan dominate the remaining term z̃>KIG>2 ∆Φ̃(xa) and the Lyapunov analysis cannotbe completed with standard techniques.

This unfortunate situation does not mean, of course, that a continuity result of this

type cannot be established. It simply reveals our inability to do it with the tools used

to analyze the ideal case.

3.5 Application to Temperature Regulation

In this subsection we design a robust PI–PBC for the temperature regulation of a class

of thermal systems—the so–called, Rapid Thermal Processes (RTP). Attention is con-

centrated on the verification of Assumption 3.3. Hence, unless otherwise indicated,

46

Assumption 3.1 is not imposed.

3.5.1 System Description

Similarly to [25, 72] we consider the following model of Rapid Thermal Processes

Ṫ = A1 [Ψ(T )−Ψ(Trad)] + A2 (T − Tconv) +Gu, (3.25)

where T ∈ Rn≥0 represents the vector of temperatures, Ψ(T ) := col(T 4i ) and Trad, Tconv ∈Rn≥0 are, respectively, the vectors of temperatures related to the radiation heat emissionfrom environment and the convection air flows. The constant matrices A1, A2 ∈ Rn×n

are Hurwitz and contain the radiation and the convection heat transfer coefficients.

Also, the entries of G ∈ Rn×m correspond to the heat transfer coefficients of the heat-ing elements. Finally, u ∈ Rm is the controlled power applied to the heating elements.Physically, considering matrix G as (3.2) means that for m heating elements there are

n−m measured points that are not directly heated by these elements.

In the model above, as in [72], it is considered that the plant is heated almost uni-

formly so that the contribution of energy from conduction is too small with respect

to the radiation transfer. Hence, the conduction heat transfer is neglected. We also

assume the steady environment conditions, i.e., the values Trad and Tconv are constant.

To simplify the notation we re–write the system (3.25) in the form

Ṫ = A1Ψ(T ) + A2T + E +Gu (3.26)

where

E := −A1Ψ(Trad)− A2Tconv.

Unlike A1, A2 and E that are highly uncertain, the input matrix G—that is defined by

the induced heat flow—can be precisely identified. The control objective is then to de-

sign a robust PI, i.e., that does not require the knowledge of the uncertain parameters,

to regulate the process around some desired temperature, which is not equal to the

open–loop equilibrium, but belongs to the assignable equilibrium set, that is,

T ? ∈{T ∈ Rn≥0 | G⊥[A1Ψ(T ) + A2T + E] = 0

}, (3.27)

where G⊥ ∈ R(n−m)×n is a full-rank left-annihilator of G.

To place the problem in the context of Proposition 3.1 we first shift the equilibrium

of the open–loop system to the origin and then proceed to verify Assumption 3.3. For,


we introduce the standard change of coordinates

x = T − T̄

where T̄ is the open–loop equilibrium that satisfies

A1Ψ(T̄ ) + A2T̄ + E = 0. (3.28)

Thus, the system (3.25) in the new coordinates takes the form (3.1) with

f(x) := A1Ψ(x+ T̄ ) + A2(x+ T̄ ) + E, (3.29)

Associated to the desired temperature T ? we define the equilibrium to be stabilised

x? := T ? − T̄ . (3.30)

3.5.2 Passivity of the thermal system.

The lemma below identifies conditions under which the system (3.25) satisfies As-

sumption 3.3 without imposing Assumption 3.1, that is, avoiding the partition of the

coordinates into actuated and un–actuated. Towards this end, the following assump-

tion is needed.

Assumption 3.4. The matrix A1 is diagonally stable [38]. That is, there exists P ∈ Rn×n,P = diag{pi} > 0 such that

PA1 + A>1 P =: −2S < 0. (3.31)

Moreover, the matrix A2 verifies

A>2 Pdiag{T 3i }+ diag{T 3i }PA2 ≤ 0. (3.32)

Conditions for diagonal stability of a matrix have been studied intensively, see [38]

for a survey. Necessary and sufficient conditions were first reported in [8]—see also

[74] for a simpler proof. For a Hurwitz matrix, a sufficient condition given in [26] is

that it is a Metzler matrix (namely, its non diagonal elements are nonnegative). Note

that due to physical nature of RTP systems the matrix A1 usually belongs to this class.

SinceA2 is Hurwitz and Ti ≥ 0, condition (3.32) is trivially satisfied ifA2 is diagonal,which is assumed in RTP models [71, 72].

48

Lemma 3.3. If Assumption 3.4 holds the vector field (3.29) satisfies Assumption 3.3

with

H(x) =n∑i=1

piφi(xi) + k (3.33)

where

φi(xi) =15(xi + T̄i)

5 −Ψi(T̄ )xi. (3.34)

and

k = −15

n∑i=1

piT̄5i .

Proof. Point (iii) of Assumption 3.3 is trivially satisfied by (3.33).

We proceed now to prove point (i). Replacing (3.34) in (3.33) and grouping terms

yields

H(x) = 15

n∑i=1

pi(xi + T̄i)5 − x>PΨ(T̄ ) + k,

Now, notice that

∇H(x) = PΦ(x),

where

Φ(x) := Ψ(x+ T̄ )−Ψ(T̄ ). (3.35)

On the other hand, from (3.28) it follows that the systems vector field may be written

as

f(x) = A1Φ(x) + A2x.

Consequently,

[∇H(x)]>f(x, θ) = Φ>(x)P [A1Φ(x) + A2x]

= −Φ>(x)SΦ(x) + Φ>(x)PA2x,

where we have used (3.31) to obtain the second identity. Now, condition (3.32) ensures

that the function h : Rn → Rn

h(x) := A>2 PΨ(x),

is monotonically decreasing [64]. That is, for all a, b ∈ Rn,

[h(a)− h(b)]>(a− b) ≤ 0.

Consequently,

Φ>(x)PA2x = [h(x+ T̄ )− h(T̄ )]>x ≤ 0

completing the proof of point (i).


To prove point (ii) we notice that

f̃(x) = A1Φ̃(x) + A2x̃,

while

∇H(x)−∇H(x?) = P Φ̃(x).

Hence, the claim is established invoking the same arguments used above and defining

Q(x) = Φ̃>(x)SΦ̃(x).

Finally, the second derivative of the functions φi(xi) yields

φ′′i (xi) = 4(xi + T̄i)3 = 4T 3i ,

which is non–negative because Ti ≥ 0. Hence, the functions φi(xi) are convex as re-quested by condition (iv) of Assumption 3.3. This completes the proof. ��

Direct application of Lemma 1 leads to the following.

Corollary 3.1. If Assumption 3.4 holds, the thermal system (3.25) defines a passive

map ũ 7→ e with storage function U(x), where

e = G>P Φ̃(x)

U(x) = H(x)− x>PΦ(x?)−H(x?) + (x?)>PΦ(x?)

3.5.3 Robust PI–PBC of the thermal system

To present the robust PI–PBC for systems verifying Assumption 3.1 we partition the

vector of temperatures into its un–actuated and actuated components

T = TuTa

, Tu :=

T1

T2...

Tn−m

, Ta :=

Tn−m+1

Tn−m+2...

Tn

,

partition P as

P = P1 0(n−m)×m0m×(n−m) D

,and do the same with the vector function Ψ(T ).

The following proposition is a consequence of Lemma 3.3 and Proposition 3.1.

50

Proposition 3.2. Consider the system (3.25) verifying Assumptions 3.1 and 3.4. Fix

any desired temperature T? verifying (3.27) and define the PI–PBC

u = −KP Ψ̃a(Ta) + z

ż = −KIΨ̃a(Ta),

and the controller gains KP and KI are given by (3.19). For all diagonal, positive defi-

nite matrices ΓP ∈ Rm×m and ΓI ∈ Rm×m all trajectories are bounded and the equilib-rium point (T, z) = (T?, u?) is globally asymptotically stable.

Proof. The proof of stability is established invoking item (i) of Proposition 1 and iden-

tifying

Φ̃a(xa)|xa=Ta−T̄a = Ψ̃a(Ta).

To prove asymptotic stability we invoke item (ii) and observe that the augmented error

signal (3.13) is given in this case by

ea =Ψ̃>(T )SG>2 D

Ψ̃(T ).Since ea verifies (3.14) and S is positive definite we conclude that Ψ̃(T (t)) → 0 andconsequently T (t)→ T ?. ��

3.5.4 Numerical Simulation:

Consider the thermal system (3.26) with

A1 =−a11 a12a21 −a22

, A2 =−α1 0

0 −α2

, G =0g

, C =c1c2

where aij ≥ 0, αi ≥ 0. Notice that the system satisfies Assumption 3.4. Then, theassignable equilibria set is

E = {T : T2 ∈ R+, −a11T 41 + a12T 42 − α1T1 + c1 = 0} (3.36)

From Proposition 3.2, the controller

ż = −KI(T 42 − (T2?)4

)u = −KP

(T 42 − (T2?)4

)+ z


300

450

600

T1,T1⋆(K)

kp=1× 10

−4→

← kp=5× 10

−6

←kp=1× 10

−8

T1⋆

300

550

800

T2,T2⋆(K)

10 (sec/div)

kp=1× 10

−4→

← kp=5× 10

−6

←kp=1× 10

−8

T2⋆

(a)

300

450

600

T1,T1⋆(K) K

I=2× 10

−5→

← KI=5× 10

−6

← KI=9× 10

−7

T1⋆

300

550

800

T2,T2⋆(K)

20 (sec/div)

KI=2× 10

−5→

← KI=5× 10

−6

← KI=9× 10

−7

T2⋆

(b)

Figure 3.1: Simulation Result showing the system response : (a) For different gains Kp letting KI =3× 10−6. (b) For different gains KI letting Kp = 6× 10−6.

where Kp = 1gΓp, KI =1gΓI and ΓP ,ΓI ∈ R+ asymptotically stabilizes the system at

T = T?. The parameter values used in the simulation where: a11 = 1 × 10−9, a12 =12 ××10

−9, a21 = 1× 10−9, a22 = 1× 10−9, α1 = 1× 10−4, α2 = 12 × 10−4, g = 1, c1 =

3, c2 = 1.7, Γp = 8 × 10−5 and ΓI = 1 × 10−5. In the simulation, the control objectiveis initially fixed at T2? = 500 K, then it is suddenly changed to T2? = 700 K. From(3.36), the corresponding values for T1? are, respectively, 430.06 K and 592.20 K. Fig.3.1 shows the simulation results. In Fig. 3.1a the response of the system when varying

control parameter Kp and letting KI = 3× 10−6 is depicted. As it can be noticed fromthe same figure, the larger is the value in Kp the faster is the convergence. In Fig. 3.1b

it is shown the response of the system when KI is varying while KP = 6× 10−6. Fromthe figure, it can be seen that large values in KI causes overshoots in the response of

T2.

Chapter 4

Energy Shaping PI of Port–Hamiltonian

Systems

An energy shaping controller for mechanical systems that does not require the solution

of partial differential equations (PDEs) has been recently proposed in [23]. In this chap-

ter we pursue this research line considering the more general case of port–Hamiltonian

(pH) systems [81].

The starting point of the design is the well–known power shaping output [55], which

defines a passive output for the pH system with storage function its energy function.

As is well–known a PI controller around this output preserves the passivity of the

closed–loop. It is then shown that, if the power shaping output is “integrable”, the

integral action of the PI is

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